Beijing-Moscow Mathematics Colloquium (online)
Abstract
The mapping class groups of oriented surfaces are important examples of groups whose properties are closely related to geometry and topology of moduli spaces, topology of 3-manifolds, automorphisms of free groups. The mapping class group of a closed oriented surface contains two important subgroups, the Torelli group, which consists of all mapping classes that act trivially on the homology of the surface, and the Johnson kernel, which is the subgroup generated by all Dehn twists about separating curves. The talk will be devoted to results on homology of these two subgroups. Namely, we will show that the k-dimensional homology group of the genus g Torelli group is not finitely generated, provided that k is in range from 2g-3 and 3g-5 (the case 3g-5 was previously known by a result of Bestvina, Bux, and Margalit), and the (2g-3)-dimensional homology group the genus g Johnson kernel is also not finitely generated. The proof is based on a detailed study of the spectral sequences associated with the actions of these groups on the complex of cycles constructed by Bestvina, Bux, and Margalit.
Bio
Prof. Alexander Gaifullin is the Correspondent member of the Russian Academy of Sciences (since 2016). He got the following honours: Prize of the President of the Russian Federation in the field os science and innovations for young scientists (2016), Prize of the Moscow Mathematical Society (2012). He is the invited speaker at the 5th European Congress of Mathematics (Krakow, 2012); plenary speaker at the 6th European Congress of Mathematics (Berlin, 2016)
